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310       XV    NORMED ALGEBRAS AND SPECTRAL THEORY

j such that (fn) is a Cauchy sequence in A, show that the sequence (o)/n) converges in
&l, and that the sequence of classes of the functions \fn\2<**fnl converges in L1.)

(f)    Let In be the interval [ , n] in R and stf \ ! the set of functions belonging to <s#
whose support is contained in IB . Show that, for each/e j/, the sequence of functions
fipin tends to /in J#. The topologies defined on d \ In by NB and by the seminorm N2
are equivalent. Deduce that the space Jf(R) is dense in ^/, and that the Banach space
A is separable.

(g)    If /and g are two functions belonging to ^, show that

g   ~     OJf       O)g

Deduce that NB(/*#) ^NB(/)NB(#) and hence that A is a commutative Banach
algebra (called Beurling's algebra}. It has no unit element.

5.   Let A be a normed algebra with no unit element. On the set A ~ A x C, define a
C-algebra structure as follows:

(x, AX*', A') = (xxf 4- A*' 4- A'x, AA'),
H(x, A) *= QJLX, /nA).

Show that (0, 1) is the unit element of A, and that \\(x, A)|| == \\x\\ 4- |A| is a norm on
A for which A is a normed algebra, and a Banach algebra if A is a Banach algebra.

2. SPECTRUM  OF AN ELEMENT OF A NORMED ALGEBRA

In this section we shall assume that A is a normed algebra with unit
element e ^ 0. For each x e A, a complex number C is said to be a regular
value for x if x  e has an inverse in A. The complex numbers C that are not
regular values for x are called the spectral values of x. The set of spectral
values of x is called the spectrum of x (in A) and is denoted by SpA(x) or
Sp(^:). When A is the algebra jSf (E) of continuous endomorphisms of a normed
space E ^ {0}, these definitions agree with those of (11,1).

Examples

(15.2.1)    For each A e C, it follows from the definition that SpA(^e) = {A}
and that SpA(jc + Ae) = SpA(x) +. A, for all x e A.

(15.2.2)    Every C e C such that x  e is a (left or right) zero-divisor in A is a
spectral value of x, but the converse is not true (Section 11.1, Problem 3).sing to the